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In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. It is named after Henri Poincaré, but it originated with Eugenio Beltrami, who used it, along with the Klein model and the Poincaré disk model (due to Riemann), to show that hyperbolic geometry was equiconsistent with Euclidean geometry. This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane. The Poincaré disk model and the half-plane model are isomorphic under a conformal mapping. This model can be generalized to model an ''n''+1 dimensional hyperbolic space by replacing the real number ''x'' by a vector in an ''n'' dimensional Euclidean vector space. ==Metric== The metric of the model on the half-plane : is given by : where ''s'' measures length along a possibly curved line. The ''straight lines'' in the hyperbolic plane (geodesics for this metric tensor, i.e. curves which minimize the distance) are represented in this model by circular arcs perpendicular to the ''x''-axis (half-circles whose origin is on the ''x''-axis) and straight vertical rays perpendicular to the ''x''-axis. In general, the ''distance'' between two points measured in this metric along such a geodesic is: : Some special cases can be simplified: :〔(【引用サイトリンク】url=http://math.stackexchange.com/q/1386297/88985 )〕 or : where ''arcosh'' is the inverse hyperbolic function area hyperbolic cosinus : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Poincaré half-plane model」の詳細全文を読む スポンサード リンク
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